Stability of integral operators on a space of homogeneous type

Abstract

Key words Stability, integral operators, spaces of homogeneous type MSC (2010) 31B10, 45P05, 46E30, 47B38, 47G10In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels K T have certain Hölder regularity and mild singularity near the diagonal. We show that given any z = 0, the L p -stability of the operator z I − T is equivalent for different 1 ≤ p ≤ ∞, where I stands for the identity operator.

Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs

“…Remark 3.2. The equivalence of unweighted stabilities for different exponents is discussed for matrices in Baskakov-Gohberg-Sjöstrand algebras, Jaffard algebras and Beurling algebras [2,39,41,47], for convolution operators [4], and for localized integral operators of non-convolution type [16,17,34,39]. For a matrix A in the Beurling algebra B r,α with 1 ≤ r ≤ ∞ and α > d G (1−1/r), Shin and Sun use the boot-strap argument in [37] to prove that reciprocal of its optimal lower unweighted stability bound for one exponent is dominated by a polynomial of reciprocal of its optimal lower unweighted stability bound for another exponent,…”

Stability for localized integral operators on weighted spaces of homogeneous type

“…The above equivalence of stability was investigated by Q. Fang and C.E. Shin [14,15] for the space (, 𝜌, 𝜇) of homogeneous type. In this paper, we will consider the weighted 𝐿 𝑝 𝑤 -stabilities for localized integral operators 𝜆𝐼 + 𝑇, 𝜆 ∈ ℂ, and by the bootstrap argument, the asymptotic estimate technique and the commutator trick, we establish their 𝐿 𝑝 𝑤 -stabilities equivalence for different exponents 1 ≤ 𝑝 < ∞ and 𝑤 ∈  𝑝 .…”

“…The above equivalence of stability was investigated by Q. Fang and C.E. Shin [14, 15] for the space $$(\mathcal {X}, \rho , \mu )$$ of homogeneous type. In this paper, we will consider the weighted $$L_w^p$$‐stabilities for localized integral operators $$\lambda I+T, \lambda \in \mathbb {C}$$, and by the bootstrap argument, the asymptotic estimate technique and the commutator trick, we establish their $$L_w^p$$‐stabilities equivalence for different exponents $$1\le p&lt; \infty$$ and $$w\in \mathcal {A}_p$$.…”